As a scaley ICS geek, I have come across this problem countless times before. Sometimes it can be attributed to a faulty video card but as you have tried another monitor which works it rules this out. Have you tried a different power cable? Chances are it is a fault with the actual mointor of which it is normally cheaper/easier to buy a new one raher than have it fixed.

Sounds like something is shorting out with something to do with the login screen etc providing the stimulus to what ever it is that is buggered. Essentially, it's fucked as the gym princess said. Warranty or new one are your options.

Often a video card problem as stated or shorting. Sometimes you can keep a monitor like this going by putting a fan at the back and blowing air through the vents at the back. It stops the bits of metal that are buggered from overheating.